A Unified Theory of Granularity, Vagueness and Approximation Thomas Bittner and Barry Smith Northwestern University NCGIA and SUNY Buffalo

Overview 1.Introduction 2.Vagueness and truth 3.Granular partitions and context 4.Vagueness and granular partitions 5.Boundaries and contexts 6.Approximation 7.Conclusions

Judging subjectSemantic theorist Partition theorist wants to determine the truth of J by using partition theory wants to determine the truth of J by using reference semantics J = ‘We will cross the boundary of Mount Everest within the next hour’ Three people and a mountain

Vagueness Where is the boundary of Everest? Boundary is subject to vagueness The boundary of Everest IS vague: broad or fuzzy boundary Vague objects and boundaries as ontological primitives Vagueness is a semantic property There is a multitude of equally good crisp candidates of reference Extend semantics: supervaluation

Supervaluation (Fine 1975) Extension of reference semantics to vagueness Takes multiplicity of candidate referents of vague names into account S = ‘X is a part of Mount Everest’ –Truth value of S is determined for all candidate referents of ‘Mount Everest’ –S is supertrue if it is true for all candidates –S is superfalse if it is true for no candidate –S is indeterminate otherwise

Vagueness and truth S = ‘We will cross the boundary of Everest within the next hour’ S is superfalse S is indeterminate S is supertrue

Vagueness and truth S = ‘We will cross the boundary of Everest within the next hour’ S is supertrue ? ? ?

Sentences vs. Judgments (Smith & Brogaard 2001) Sentence: ‘There is no beer in the glass.’ Drunkard: Hygiene inspector: Judgments = Sentence + Context (super) true The glass does not contain (drinkable amounts of) beer (super) false The glass contains tiny amounts of beer, microbes, mold, …

Granular partitions a formally tractable proxy for the notion of context

Theory of granular partitions There is a projective relation between cognitive subjects and reality Major assumptions: Humans ‘see’ reality through a grid The ‘grid’ is usually not regular and raster shaped

Projection of cells … Wyoming Idaho Montana … Cell structure North America Projection

no counties no county boundaries Part of the surface of the Earth photographed from space Projection establishes fiat boundaries Cell structure Map = Representation of cell structure County boundaries in reality P

Partitions and context J = (‘There is no beer in the glass’, Partition) Glass Beer Glass Beer probe Cell ‘Beer’ does project Cell ‘Beer’ does not project J is true in this context J is false in this context

Judgments about mereological structure J = (‘X is part of Y’, Pt) = true Y X UV Labeling of names in S onto cells in Pt projection

Vagueness and granular partitions

Crisp and vague projection … Montana … crisp Himalayas Everest vague P1P1 PnPn Vague reference is always reference to fiat boundaries!

Vague judgments about mereological structure J = (‘X V is part of Y’, Pt V ) = supertrue Y X Labeling of names in S onto cells in Pt P1P1 PnPn

Vagueness and truth J = (‘We will cross the boundary of Everest within the next hour’, Pt) ? ? ? Whether or not indeterminacy can arise depends on the projection of the boundaries!

Boundaries and contexts

We distinguish: contexts in which our use of a vague term brings: 1.a single crisp fiat boundary 2.a multiplicity of crisp fiat boundaries into existence

The single crisp boundary case J = (‘This is the boundary of Mount Everest’, Pt) The judging subject must have the authority (the partitioning power) to impose this boundary e.g., she is a member of some government agency Vagueness is resolved. J has a determinate truth value

The multiple boundary case The subject (restaurant owner) judges: J = (‘The boundary of the smoking zone goes here’, Pt) while vaguely pointing across the room. Vague projection brings a multitude of boundary candidates into existence Truth-value indeterminacy can potentially arise To show: naturally occurring contexts are such that truth-value indeterminacy does not arise.

The multiple boundary case Claim: The judgment can be uttered only in contexts (1) Where it is precise enough to be (super)true (2) but: not precise enough for indeterminacy to arise The subject (restaurant owner) judges: J = (‘The boundary of the smoking zone goes here’, Pt) while vaguely pointing across the room.

The multiple boundary case Context 1: To advise the staff where to put the ashtrays The projection must be just precise enough to determine on which table to put an ashtray The subject (restaurant owner) judges: J = (‘The boundary of the smoking zone goes here’, Pt) while vaguely pointing across the room. No truth-value indeterminacy Context 2: To describe where nicotine molecules are truth-value indeterminacy can potentially occur But: nobody can seriously utter such a judgment in naturally occurring contexts

Approximation; or how to make vague reference in a determinate fashion

Boundaries limiting vagueness S = ‘We will cross the boundary of Everest within the next hour’ in one hour, interior boundary Exterior b. now candidate i candidate k direction of travel core Where-the-boundary-candidates are Two partitions: (1) a vague partition Carving out candidate referents for the vague name ‘Everest’ (2) a partition projecting along the way ahead Limits admissible candidate referents for ‘Everest’

Approximating judgments S = ‘We will cross the boundary of Everest within the next hour’ J = (S, Pt V, Pt R ) in one hour, interior boundary Exterior b. now candidate i candidate k direction of travel core Where-the-boundary- candidates are Truth of J depends on the relationships between Pt V and Pt R

Truth of approximating judgments An approximating judgment J = (S, Pt V, Pt R ) is: Supertrue: all candidate referents projected onto by Pt V are within the limits given by Pt R Superfalse: no candidate referent projected onto by Pt V is within the limits given by Pt R Indeterminate: some candidate referents projected onto by Pt V are within the limits given by Pt R and others are not.

Truth-value indeterminacy of approximating judgments … ? … does not actually occur in naturally occurring contexts

Truth value indeterminacy ?? Why can ‘We will cross the boundary of Everest within the next hour’ not be judged in these contexts ? ? ? Judger has the freedom to choose appropriate delimiting boundaries. Why should she use ridiculous ones which do not make sense ? Why should she use ones subject to indeterminacy ? ?

Higher order vagueness Boundaries that delimit vagueness of reference What if these boundaries are subject to vagueness themselves? Higher-order vagueness S = ‘We will cross the boundary of Everest within the next hour or so’

Higher order vagueness To show: Higher order vagueness does not cause truth-value indeterminacy in naturally occurring contexts S = ‘We will cross the boundary of Everest within the next hour or so’ (1)those which re-use existing boundaries (2) those which create new fiat boundaries Two classes of contexts:

Re-using existing boundaries J = (‘The area of bad weather extends over parts of Wyoming, Montana, Idaho, and Utah’, Pt V, Pt R ) The re-used boundaries are crisp. No truth value indeterminacy

Create new fiat boundaries S = ‘We will cross the boundary of Everest within the next hour or so’ Multiplicity of candidate referents Judging subject must choose limiting boundaries much crisper than the degree of vagueness they limit

Conclusions Theory of granular partitions provides a tool to understand granularity, vagueness, indeterminacy and the relationships between them Context is critical when analyzing truth-values of judgments In naturally occurring contexts truth-value indeterminacy does not occur Formalism – see paper Partition-theoretic solution to the Sorites paradoxes – see paper